Mathematical modeling supports substantial mouse neural progenitor cell death
© McConnell et al. 2009
Received: 18 March 2009
Accepted: 14 July 2009
Published: 14 July 2009
Existing quantitative models of mouse cerebral cortical development are not fully constrained by experimental data.
Here, we use simple difference equations to model neural progenitor cell fate decisions, incorporating intermediate progenitor cells and initially low rates of neural progenitor cell death. Also, we conduct a sensitivity analysis to investigate possible uncertainty in the fraction of cells that divide, differentiate, and die at each cell cycle.
We demonstrate that uniformly low-level neural progenitor cell death, as concluded in previous models, is incompatible with normal mouse cortical development. Levels of neural progenitor cell death up to and exceeding 50% are compatible with normal cortical development and may operate to prevent forebrain overgrowth as observed following cell death attenuation, as occurs in caspase 3-null mutant mice.
intermediate progenitor cell
in situ end-labeling plus
neural progenitor/precursor cell
programmed cell death
terminal dUTP nick-end labeling
Essentially every excitatory neuron in the cerebral cortex is born from a heterogeneous pool of mitotic cells (referred to collectively as neural progenitor/precursor cells (NPCs)) in the embryonic ventricular zone (VZ) [1–4]. During a 'neurogenic interval' in mouse – commencing at embryonic day (E)10 in the rostro-medial cortex and concluding at E18 in the caudo-lateral cortex [5–7] – the founding NPC population expands through proliferative divisions until it is exhausted by terminal differentiation and programmed cell death (PCD). NPC proliferation must be balanced with the operation of PCD to produce a sufficient, but not supernumerary, neuronal population. Understanding of the cellular and genetic mechanisms controlling the size of the cerebral cortex, among the most notable distinctions of the brain's evolution [8, 9], could benefit from an accurate quantitative model of the fate decisions made by NPCs.
Initial models of mouse NPC fate decisions are insufficiently constrained because it was assumed that cell divisions outside of the VZ do not contribute cortical neurons ; however, it is now clear that non-VZ mitoses contribute significant numbers of cortical neurons [11–13]. Earlier models also lacked direct measurement of an important parameter, the founding NPC population size, instead relying on two related but less direct measurements : changes in the size of the VZ (from which the number of NPCs is extrapolated) and the fraction of cells emigrating from the VZ (from which VZ neuronal output is calculated). Direct measurement of the size of the founding NPC pool [15, 16] provides a more accurate description than extrapolation and addresses a limitation of early models. These data are employed to constrain new models as pursued here.
A direct quantitative implication of non-VZ neuronal production is that additional PCD will be required to offset additional neuronal production. Reports of previous models contend that NPC death is an insignificant component of neurodevelopment [17–19] despite empirical data that are consistent with significant NPC death. Data supporting much higher levels of NPC death than proposed in prior models were first reported using a sensitive DNA end-labeling technique, 'in situ end-labeling plus' (ISEL+), and ligation-mediated PCR [20–24]. Additional support came from recent analyses of NPC progeny, marked using a genomically encoded lineage tracer; here, the progeny clone size was found to diminish markedly at E14 . Perhaps most compelling, deletion of numerous pro-cell death genes, including those encoding caspase 3 , caspase 9 , APAF1 , Bax, Bak , and Pten , as well as novel molecules like ephrins , all lead to brain overgrowth phenotypes. Conversely and consistently, null mutations in pro-survival genes (for example, those encoding Bcl-x , Survivin  and Mcl-1 ) lead to smaller brains. Given this large body of empirical evidence, new models should account for the more extensive operation of PCD during cortical neurogenesis.
Here we report new quantitative models that incorporate new data and are consistent with cortical PCD empirical evidence. In particular, with respect to total neuronal production, we demonstrate a clear requirement for substantial NPC death during mouse cerebral cortical development. These models further provide a quantitative explanation of neurodevelopmental cortical overgrowth phenotypes produced by PCD attenuation as observed in caspase 3-null mutant mice.
Simple models of NPC fate decisions require intermediate levels of cell death
Model parameters and constraints
Founding NPC population
5 to 6 × 105
NPC CC number
1 to 11
Fraction of NPCs that become newly
q 1 = 0.005; q 2 = 0.04;
q 3 = 0.09;q 4 = 0.14;
q 5 = 0.21;q 6 = 0.31;
q 7 = 0.42; q 8 = 0.54;
q 9 = 0.69;q 10 = 0.84;
q 11 = 1.0
Fraction of NPCs that die at i th CC
5% to >50%
After i th CC, number of new:
Dying daughter cells
Total neurons produced from NPCs
1 × 107 to 2.72 × 107
10 to 16 × 106
15 to 30% of cortical neurons
Post-natal cell death
30 to 50% of cortical neurons
Fraction of IPCs at each CC
q' 1 = 0; q' 2 = 0; q' 3 = 0;
q' 4 = 0; q' 5 = 0.12;
q' 6 = 0.12; q' 7 = 0.16;
q' 8 = 0.16; q' 9 = 0.54;
q' 10 = 0.54; q' 11 = 0.54
The total number of neurons in the adult mouse cerebral cortex has been estimated at between 1.0 × 107 and 1.6 × 107 (Table 1 and references therein), and we use this estimate to constrain simulations of VZ output. Approximately 15 to 30% of the estimated total neuronal population consists of inhibitory interneurons. In addition, the estimate of 1.0 to 1.6 × 107 neurons includes cell loss that is due to 30 to 50% post-mitotic (non-VZ) cell death (Table 1 and references therein). Taking these figures into consideration, we can determine a range for plausible VZ output as follows. First, a lower bound on VZ output corresponds to taking the low estimate of 1.0 × 107 neurons and assuming that 30% of these are interneurons and that there was only 30% post-mitotic death; that is, a lower bound on VZ output is ((1.0 × 107) × (1 – 0.3))/(1 – 0.3) = 1.0 × 107. Similarly, an upper bound on VZ output follows from taking 1.6 × 107 neurons and assuming that 15% are interneurons and that 50% post-mitotic cell death occurred; thus, a plausible upper bound on VZ output is ((1.6 × 107) × (1 – 0.15))/(1 – 0.5)) = 2.72 × 107.
Therefore, data suggest that an accurate quantitative model of mouse cerebral cortical neurogenesis should yield a plausible range of VZ output between 1.0 × 107 and 2.72 × 107. However, the published estimate of 140 progeny per founding NPC  corresponds to 7.7 × 107 NPC progeny produced by 5.5 × 105 founding NPCs. NPC death was presumed negligible when calculating this progeny-per-NPC estimate and it cannot be reconciled with the maximum plausible VZ output.
Caviness and colleagues  calculated that a founding population of 2.5 × 105 NPCs was compatible with their model. The plausibility window when P 0 = 2.5 × 105 accommodates between 5 and 17% NPC death for model DG1 and between 4 and 15% NPC death for model DG2 (data not shown). These calculations suggest that a significant reduction of the founding NPC pool could be consistent with lower levels of cell death during development; however, this explanation is not consistent with two independent measurements of the founding NPC pool size [15, 16].
Our models allow us to view the range of plausible VZ output as a function of PCD over the course of 11 CCs, and we refer to this correspondence as the 'plausibility window.' For example, as shown in Figure 1C,D, a plausible range of VZ output is observed when 13% <d i < 26% for all i. Because of the delayed depletion of the proliferative population, the corresponding range of plausible d i values in model DG1 is broader than that calculated using model DG2 (that is, 15 to 26% versus 13 to 23%, respectively). Neither model DG1 or DG2 matches the experimental measurements of either 5% NPC death using TUNEL or 50% NPC death using ISEL+; however, these calculations do demonstrate that 5% NPC death is too low to calculate VZ output adequately in the normal mouse brain.
Sensitivity analysis of model DG1
Measurements of CC duration together with the fraction of NPCs that differentiate at experimentally defined ages permits extrapolation of model parameter q i for each i . Sensitivity analysis  provides a means of determining the relationship between inherent uncertainty in these estimates of q i and uncertainty in simulated VZ output.
Representative sensitivity analysis
Intermediate progenitor cells alter the plausibility window
NPC mitoses occur on the ventricular surface, but 'non-surface' mitoses are also observed in the developing cortex. Initially, these cell divisions were erroneously considered non-neuronogenic , but proliferative intermediate progenitor cells (IPCs) – daughter cells of NPCs that have migrated to the subventricular zone and intermediate zones, and are immunoreactive for the transcription factor Tbr2 – are now known to contribute additional excitatory neurons to the mouse cerebral cortex [11–13].
Cortical development with 5% NPC death at early CCs requires more than 50% NPC death at later CCs
A primary objection to the idea of substantial NPC death is the intuition that ≥ 50% NPC death throughout neurodevelopment would preclude expansion of the NPC pool. However, the level of NPC death need not be constant at each CC. For example, ISEL+ labels approximately 5% of cells at E10 and the percentage increases significantly thereafter [20, 35]. Moreover, NPC lineage analyses using a genomically encoded marker found that clone size increased during early development but then diminished after E14 , suggesting significant cell death with corpse elimination at later, but not earlier, CCs. These empirical observations are also consistent with increased model sensitivity to the levels of death at earlier CCs (Table 2). Together, these data suggest that early 'expansion' CCs occur; initially, low PCD levels – for example, setting d i = 0.05 for 1 ≤ i ≤ 4 provides for four expansion CCs – further constrain VZ output. Here we calculate the corresponding amount of NPC death required during later CCs for plausible VZ output.
We note here that inclusion of four or five expansion CCs makes model DG1 and model DG2 readily distinguishable from one another. This distinction is a consequence of the differential involvement of cell death when expansion CCs begin and end, and illustrates some additional sensitivity to the size of the founding NPC population. As described above, during the first CC in model DG2 the founding population is reduced before mitosis; therefore, for uniformly constant cell death, model DG2 with initial population P 0 is equivalent to model DG1 with initial population d 1 P 0. Although this distinction leads to only subtle differences in most simulations, when considering the relationship between expansion CCs and plausible levels of NPC death, model DG1 with four or five expansion CCs illustrates additional potential for biological variation.
Early expansion CCs and IPCs together are most compatible with experimental data and further support high levels of NPC death
Cell death amongst NPCs is a prominent feature of neurogenesis in other regions of the nervous system (for example, retina; reviewed in [38, 39]); however, the amount of NPC death during mouse cerebral cortical development is debated [40, 41]. Employing published cell counts to estimate ranges of neuronal population size in the mouse cerebral cortex, we calculated a plausible range of VZ output for normal mouse neurodevelopment. Models DG1 and DG2 use simple difference equations to capture essential features of previous probabilistic models. Model DG1 with three to five early expansion CCs, a pool of IPCs, and NPC death near 50% incorporates the most experimentally observed constraints and is consistent with NPC death levels as observed in ISEL+ analyses [22, 42, 43].
Comparison with contemporary models
Apart from initial modeling by Takahashi, Caviness, and colleagues (referenced throughout), Gohlke, Faustman, and colleagues [17, 44] have used a Kolmogorov forward equation to compute the probability distribution for mouse VZ output explicitly, as a continuous-time Markov chain. In contrast, we estimate this same probability distribution by repeatedly simulating a deterministic difference equation with random perturbations taken from assumed distributions. This allows us to illustrate how significant variation in q i and d i fractions, over the course of 11 CCs, can occur and still yield plausible VZ output (Figure 2B). In population measurements using stereological counting, variation at the level of subpopulations of NPCs would probably go unnoticed, yet this may be an important feature of cerebral cortical development. While such variation is implicit in the Kolmogorov forward equation approach, Monte Carlo allows us to view sample trajectories that illustrate this variation.
The Gohlke models also observe that the original Takahashi models lead to VZ output that exceeds experimental counts of cortical neurons by at least threefold . However, these authors use two additional parameters (in addition to low-level NPC death) to reduce cortical neuron production to plausible levels: a diminished growth fraction insofar as not all VZ cells are progenitor cells and a clearance time for dying NPCs. Many NPCs co-label for bromodeoxyuridine and ISEL+ [20, 24, 42]; therefore, Gohlke and colleagues have likely recast some additional NPC death as a 'diminished' growth fraction. These parameters have the effect of reducing the proliferative population from which any given q fraction is taken. Consequently, the overproduction observed using the Takahashi model is limited and plausible levels of VZ output are obtained from the Gohlke mouse model [17, 44].
The Gohlke mouse model is reportedly compatible with levels of NPC death up to 24% . This value is in good agreement with 13 to 26% NPC death calculated using models DG1 and DG2 (Figure 1). Despite this, Gohlke and colleagues [17, 45] report subsequent model analysis using NPC death rates at or near 0% NPC death. The contrasting higher levels of NPC death required in Gohlke models of primate cortical neurogenesis , relative to murine cortical neurogenesis, may simply reflect an underestimation of mouse NPC death.
TUNEL underestimates NPC death
A strong implication of all modeling experiments presented here is that TUNEL significantly underestimates NPC death during mouse cerebral cortical development. This reflects technical differences in sensitivity between the two procedures, with ISEL+ being approximately ten times more sensitive than the originally reported TUNEL technique [21, 22]. ISEL+ detects more dying cells not only amongst NPCs, but also in other tissues like the thymus and small intestinal villus . Consistent with a tenfold reduced sensitivity relative to ISEL+, TUNEL detects as few as one-tenth of the dying NPCs (5% versus 50%).
Consideration of the caspase 3-deficient phenotype
In order to reconcile the approximately 5% NPC death proposed by prior models  with the forebrain overgrowth phenotype observed in caspase 3-deficient mice, it has been proposed that NPC death may occur normally in a small population of neuroepithelial stem cells or radial glia at an early age (<E12) . The claim is that additional survival of a few such cells could underlie forebrain overgrowth in caspase 3-deficient mice because each of these individual cells might ultimately give rise to many neurons (approximately 140 according to estimates from concurrent models). However, this notion is inconsistent with experimental data demonstrating marked overgrowth of NPCs in caspase 3-null embryos by E12, before large numbers of NPC progeny emigrate to the cortex [26, 42].
Our models are consistent with changes in NPC and neuronal populations that have been observed following experimental attenuation of PCD. For example, a 20% reduction in NPC death from the midpoint of any plausibility window corresponds to VZ output that exceeds the upper bound for that plausibility window. This is strikingly consistent with a 30% reduction in ISEL+ labeling of NPCs observed in caspase 3-deficient mice , where exceptional forebrain overgrowth is observed. We suggest that an in vivo correlate of 'exceeding the plausibility window' is forebrain overgrowth.
Predictions derived from observing the plausibility window
The breadth of the plausibility window provides a comparative measure of model robustness with respect to viable levels of NPC death. In some scenarios the slope of VZ output is steep (Figure 1D), and a 10% difference in NPC death leads to marked differences in VZ output, while a similar 10% change has little impact on VZ output in models where the slope is less steep (for example, Figure 4E). Notably, early expansion CCs (Figure 4), rather than the size of the founding NPC population or additional IPC progeny (compare Figures 2 and 3), significantly extend the range of NPC death that is compatible with plausible VZ output. Although modeled during the first three to five CCs here because of experimental evidence for pre-E12 expansion CCs [20, 25], transient low-level NPC death could theoretically operate during any CC. It is tempting to speculate that high levels of NPC death at later CCs follow from low levels of NPC death during early expansion CCs. A similar process has been reported in embryonic stem cells earlier in development [47, 48] and may be related to a permissive decatenation checkpoint observed in NPCs .
One might predict that other perturbations leading to transient high levels of NPC death at early CCs could lead to low level NPC death at later CCs. In this scenario, developmental accommodation of atypical NPC death might occur at the level of stem cell niches [49, 50], providing sufficient, but not necessarily 'normal,' VZ output. Perhaps local control of NPC death could insulate cerebral cortical development against genetic differences and chemical or environmental insults. Given genetic diversity among NPCs, produced in part by chromosomal aneuploidy  and retrotransposition , differences in NPC death amongst individuals suggests selection and/or survival mechanisms that influence the mosaic composition of an individual's cerebral cortex.
Models DG1 and DG2 resolve discrepancies existing between previous models and experimental data; furthermore, these models provide a quantitative account for qualitative differences observed during PCD-attenuated cerebral cortical development. This theoretical framework should motivate additional experimental investigation of expansion CCs and reinterpretation of other cortical development phenotypes that measured PCD among NPCs using only TUNEL staining.
Materials and methods
The order in which NPC death, differentiation, and proliferation are imposed alters VZ output, so it is natural to consider two related models. For the first model, DG1, the number of generated neurons at the i th CC is Q i = 2q i P i-1, where P i-1 is given by P i = (1 - d i )(1 - q i )2P i-1. Here, (1 - q i )2P i-1 is the number of non-emigrating daughter cells after the i th division and d i (1 - q i )2P i-1 of these die, leaving P i (1 - d i )(1 - q i )2P i-1 to divide again. Alternatively, imposing death before imposing differentiation gives Q i = q i (1 - d i ) 2P i-1 while, again, P i (1 - d i )(1 - q i )2P i-1 many NPCs go through the i th CC. We refer to this model as DG2. In each of these two related models, the total VZ output over 11 NPC CCs is . Viewing VZ = VZ(q, d) as a function of all q i and d i – that is, q = (q i ) and d = (d i ) are 11-dimensional vectors – allows us to explore how total VZ output is sensitive to variation in q i and d i .
where Var [VZ(q, d)] is the variance in VZ output that arises when all q i and d i are randomized (for example, 10,000 runs) and where Var [E[VZ(q, d|q k )]], for example, is the variance in expected VZ output that arises when, for some fixed k, all q i with i ≠ k are allowed to vary. That is, to compute Var [E[VZ(q, d|q k )]], q k is fixed repeatedly while for all i ≠ k and for all i = 1, 2,..., 11. Then, the average (expected) VZ output is taken, and this entire process is repeated (for example, approximately 100 times) to determine the variance of such an average.
Calculations were performed using MatLab version 7.7.0 (Mathworks, Natick, MA, USA) as detailed in the text. Figures were prepared using Illustrator and Photoshop (Adobe Systems Inc., San Jose, CA, USA). The MatLab scripts are available as additional files 1 and 2.
We are very grateful to Drs S Rehen, D Kaushal, M Kingsbury, A Yang, L Cai, L Chao, T Haydar, and P Yeh for helpful discussions. This work was supported by a NIGMS pharmacology training grant (MM), NSF/EPSCoR grant no. EPS 0447660 (HM), and MH51699 (JC).
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